The equation is used as a toy model for the 3D [[Euler equation]] and [[Navier-Stokes]]. The main question is to determine whether the Cauchy problem is well posed in the classical sense. In the inviscid case, it is a major open problem as well as in the supercritical diffusive case when $s<1/2$. It is believed that inviscid SQG equation presents a similar difficulty as 3D Euler equation in spite of being a scalar model in two dimensions {{Citation needed}}. The same comparison can be made between the supercritical SQG equation and [[Navier-Stokes]].

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The equation is used as a toy model for the 3D Euler equation and Navier-Stokes. The main question is to determine whether the Cauchy problem is well posed in the classical sense. In the inviscid case, it is a major open problem as well as in the supercritical diffusive case when $s<1/2$. It is believed that inviscid SQG equation presents a similar difficulty as 3D Euler equation in spite of being a scalar model in two dimensions <ref name="CMT"/>. The same comparison can be made between the supercritical SQG equation and Navier-Stokes.

The key feature of the model is that the drift $u$ is a divergence free vector field related to the solution $\theta$ by a zeroth order singular integral operator.

The key feature of the model is that the drift $u$ is a divergence free vector field related to the solution $\theta$ by a zeroth order singular integral operator.

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For the diffusive case, the well posedness of the equation follows from perturbative techniques in the subcritical case ($s>1/2$). In the critical case the proof is more delicate and can be shown using three essentially different methods. In the sueprcritical regime ($s<1/2$) only partial results are known.

For the diffusive case, the well posedness of the equation follows from perturbative techniques in the subcritical case ($s>1/2$). In the critical case the proof is more delicate and can be shown using three essentially different methods. In the sueprcritical regime ($s<1/2$) only partial results are known.

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Global weak solutions, as well as classical solutions locally in time, are known to exist globally for the full range of $s \in [0,1]$.

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Global weak solutions, as well as classical solutions locally in time, are known to exist globally for the full range of $s \in [0,1]$ <ref name="R" />.

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If $\theta$ solves the equation, so does the rescaled solution $\theta_r(t,x) = r^{2s-1} \theta(r^{2s} t,rx)$.

If $\theta$ solves the equation, so does the rescaled solution $\theta_r(t,x) = r^{2s-1} \theta(r^{2s} t,rx)$.

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The $L^\infty$ norm is invariant by the scaling of the equation if $s=1/2$. This observation makes $s=1/2$ the critical exponent for the equation. For smaller values of $s$, the diffusion is stronger than the drift in small scales and the equation is well posed. For larger values of $s$, the drift might be dominant at small scales.

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The $L^\infty$ norm is invariant by the scaling of the equation if $s=1/2$. This observation makes $s=1/2$ the critical exponent for the strongest a priori estimate available. For larger values of $s$, the diffusion dominates the drift in small scales and the equation is well posed. For larger values of $s$, the drift might be dominant at small scales.

== Well posedness results ==

== Well posedness results ==

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=== Sub-critical case: $s>1/2$ ===

=== Sub-critical case: $s>1/2$ ===

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The equation is well posed globally. The proof can be done with several methods using only soft functional analysis or Fourier analysis.

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The equation is well posed globally. The proof can be done with [[perturbation methods]] using only soft functional analysis or Fourier analysis.

=== Critical case: $s=1/2$ ===

=== Critical case: $s=1/2$ ===

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The equation is well posed globally. There are three known proofs.

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The equation is well posed globally. There are four known proofs.

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* '''Evolution of a modulus of continuity''': An explicit modulus of continuity which is comparable to Lipschitz in small scales but growth logarithmically in large scales is shown to be preserved by the flow. The method is vaguely comparable to [[Ishii-Lions]].

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* '''Evolution of a modulus of continuity''' <ref name="KNV"/>: An explicit modulus of continuity which is comparable to Lipschitz in small scales but growth logarithmically in large scales is shown to be preserved by the flow. The method is vaguely comparable to [[Ishii-Lions]].

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* '''De Giorgi approach''': From the $L^\infty$ modulus of continuity, it is concluded that $u$ stays bounded in $BMO$. A variation to the parabolic [[De Giorgi-Nash-Moser]] can be carried out to obtain Holder continuity of $\theta$.

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* '''De Giorgi approach''' <ref name="CV"/>: From the $L^\infty$ modulus of continuity, it is concluded that $u$ stays bounded in $BMO$. A variation to the parabolic [[De Giorgi-Nash-Moser]] can be carried out to obtain Holder continuity of $\theta$. The result does not use the relations $u = R^\perp \theta$, but only that $u$ is a divergence-free vector field in ''BMO''. Therefore, it is actually a regularity result for arbitrary [[drift-diffusion equations]].

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* '''Dual flow method''': Also from the information that $u$ is $BMO$ and divergence free, it can be shown that the solution $\theta$ becomes Holder continuous by studying the dual flow and characterizing Holder functions in terms of how they integrate against simple test functions.

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* '''Dual flow method''' <ref name="KN"/>: Also from the information that $u$ is $BMO$ and divergence free, it can be shown that the solution $\theta$ becomes Holder continuous by studying the dual flow and characterizing Holder functions in terms of how they integrate against simple test functions. This is a regularity result for general [[drift-diffusion equations]] as well.

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* '''Nonlinear maximum principle''' <ref name="ConstVicol"/>: By studying the evolution of $|\nabla \theta|^2$ and using a [[nonlinear lower bound on the fractional Laplacian]] when evaluated at extrema, one may prove that a solution of SQG which has only (sufficiently) small jumps, then it is in fact smooth. By looking at the evolution of finite differences $\delta_h \theta(x) = \theta(x+h) - \theta(x)$, one may measure the evolution of these small jumps, and prove that if the initial data has only small jumps, then so does the corresponding solution of SQG. Combined, the above statements imply global regularity.

The equation is used as a toy model for the 3D Euler equation and Navier-Stokes. The main question is to determine whether the Cauchy problem is well posed in the classical sense. In the inviscid case, it is a major open problem as well as in the supercritical diffusive case when $s<1/2$. It is believed that inviscid SQG equation presents a similar difficulty as 3D Euler equation in spite of being a scalar model in two dimensions [1]. The same comparison can be made between the supercritical SQG equation and Navier-Stokes.

The key feature of the model is that the drift $u$ is a divergence free vector field related to the solution $\theta$ by a zeroth order singular integral operator.

For the diffusive case, the well posedness of the equation follows from perturbative techniques in the subcritical case ($s>1/2$). In the critical case the proof is more delicate and can be shown using three essentially different methods. In the sueprcritical regime ($s<1/2$) only partial results are known.

Global weak solutions, as well as classical solutions locally in time, are known to exist globally for the full range of $s \in [0,1]$ [2].

Scaling and criticality

If $\theta$ solves the equation, so does the rescaled solution $\theta_r(t,x) = r^{2s-1} \theta(r^{2s} t,rx)$.

The $L^\infty$ norm is invariant by the scaling of the equation if $s=1/2$. This observation makes $s=1/2$ the critical exponent for the strongest a priori estimate available. For larger values of $s$, the diffusion dominates the drift in small scales and the equation is well posed. For larger values of $s$, the drift might be dominant at small scales.

Well posedness results

Sub-critical case: $s>1/2$

The equation is well posed globally. The proof can be done with perturbation methods using only soft functional analysis or Fourier analysis.

Critical case: $s=1/2$

The equation is well posed globally. There are four known proofs.

Evolution of a modulus of continuity[3]: An explicit modulus of continuity which is comparable to Lipschitz in small scales but growth logarithmically in large scales is shown to be preserved by the flow. The method is vaguely comparable to Ishii-Lions.

De Giorgi approach[4]: From the $L^\infty$ modulus of continuity, it is concluded that $u$ stays bounded in $BMO$. A variation to the parabolic De Giorgi-Nash-Moser can be carried out to obtain Holder continuity of $\theta$. The result does not use the relations $u = R^\perp \theta$, but only that $u$ is a divergence-free vector field in BMO. Therefore, it is actually a regularity result for arbitrary drift-diffusion equations.

Dual flow method[5]: Also from the information that $u$ is $BMO$ and divergence free, it can be shown that the solution $\theta$ becomes Holder continuous by studying the dual flow and characterizing Holder functions in terms of how they integrate against simple test functions. This is a regularity result for general drift-diffusion equations as well.

Nonlinear maximum principle[6]: By studying the evolution of $|\nabla \theta|^2$ and using a nonlinear lower bound on the fractional Laplacian when evaluated at extrema, one may prove that a solution of SQG which has only (sufficiently) small jumps, then it is in fact smooth. By looking at the evolution of finite differences $\delta_h \theta(x) = \theta(x+h) - \theta(x)$, one may measure the evolution of these small jumps, and prove that if the initial data has only small jumps, then so does the corresponding solution of SQG. Combined, the above statements imply global regularity.

Supercritical case: $s<1/2$

The global well posedness of the equation is an open problem. Some partial results are known: